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Why are there so many maximum and comparison principles in the study of partial differential equations? It is scary to try and learn them because there are hundreds of them. Does every type of domain require a new principle?

hopo2
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  • Examples? There are only two in my humble math world: weak and strong. – timur Aug 23 '12 at 20:47
  • @timur See this book: http://books.google.co.uk/books?id=xNOkdVVdmRAC&pg=PA276&lpg=PA276&dq=strong+comparison+principle&source=bl&ots=JEc7i2PLJe&sig=H5GkOB8hpgLxomEjOBthal9Oi0U&hl=en&sa=X&ei=0ZU2ULm0DIWW0QWc34DIDw&ved=0CDwQ6AEwAg#v=onepage&q=strong%20comparison%20principle&f=false

    Also there's a whole book Max. Principles by Protter, I believe..

     – hopo2 Aug 23 '12 at 20:48
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    I don't see any problem. If you are not doing research on maximum principles, then just learn the ones you need as you go. If you do research on maximum principles, then that means you are interested in it, so no complaints. – timur Aug 23 '12 at 20:52
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    @hopo2 To avoid having your question closed, I would suggest that you edit it to be more substantive. For instance, "Does every type of domain require a new maximum/comparison principle?" sounds like a good question, everything else weakens your question and discourages others from answering it. – mboratko Aug 23 '12 at 21:07
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    The book by Pucci and Serrin is called "The Maximum Principle", but switches to plural in the first sentence. I think it's not so much about different types of domains as about different types of equations: parabolic vs elliptic, divergence form vs non-divergence, etc, etc. –  Aug 23 '12 at 21:13
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    One could apply this same argument to mathematics as a whole. I don't see the problem. – Qiaochu Yuan Aug 23 '12 at 21:59
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    Just in case my earlier comment gave the wrong impression that you just have to accept the state it is in now, what I meant was that you try organize the theory as best as you can, but still you cannot reduce everything into a single sentence. There is some inherent complexity in any subject. What is more, it is not as chaotic as you seem to think. – timur Aug 24 '12 at 14:12

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The most rough way I can say the "maximum principle" for elliptic PDE is as follows: Take a second order elliptic PDE $F(D^2u,Du,u,x) = 0$. (Ellipticity in this context means that if $M \geq N$ as matrices, then $F(M,p,z,x) \geq F(N,p,z,x)$). If I can find a family of strict subsolutions $v(t)$ that $v(0)$ is below $u$ everywhere and $v(t) \leq u$ on the boundary for all $t$, then $v(t)$ must always lie below $u$ on the interior. Indeed, if not then for some first time a $v(a)$ would touch $u$ by below at some point $x$ on the interior, giving the contradiction

$$0 < F(D^2v(x),Dv(x),v(x),x) \leq F(D^2u(x),Du(x),u(x),x) = 0.$$

Most of the maximum principles I know are special cases of this. If constants are solutions, then the minimum must occur on the boundary by sliding horizontal planes from below. (Actually, since planes aren't strict subsolutions we need to "convexify" planes a bit and assume something about the dependence of $F$ on $Du$ so that they become strict subsolutions and still touch by below on the interior, giving the desired contradiction).

Even the Hopf lemma boils down to this idea; you produce a radially symmetric subsolution that intersects horizontal planes at an angle. The assumption that $F$ decreases in the $u$ variable is usually imperative to the maximum principle because we want to create a family of subsolutions by taking the original one and sliding down.

I hope this helps!

Calvin McPhail-Snyder
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Connor
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